Journal of Mathematics Research; Vol. 10, No. 4; August 2018 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education On the Trapped Surface Characterization of the Black Hole Region in Kerr Spacetime Mohammed Kumah1 & Francis T. Oduro2 1 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana 2 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Correspondence: Mohammed Kumah, PMB, Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. E-mail: [email protected] Received: December 22, 2017 Accepted: January 8, 2018 Online Published: May 28, 2018 doi:10.5539/jmr.v10n4p24 URL: https://doi.org/10.5539/jmr.v10n4p24 Abstract Black holes are classically characterized by event horizon which is the boundary of the region from which particles or photons can escape to infinity in the future direction. Unfortunately this characterization is a global concept as the knowledge of the whole spacetime is needed in order to locate a black hole region and the event horizon. It is therefore important to recognize black holes locally; this has motivated the need to use local approach to characterize black holes. Specifically, we apply covariant divergence and Gausss divergence theorems to compute the divergences and the fluxes of appropriate null vectors in the Kerr spacetime to actually determine the existence of trapped and marginally trapped surfaces in its black hole region. Keywords: Trapped surfaces, black holes, event horizons 1. Introduction One of the most striking results of General Relativity is its prediction of black holes which are spacetime regions from which no signal can be seen by an observer far from the matter sources (Frolov and Zelnikov, 2011). These Black holes are formed through the gravitational collapse of sufficiently massive objects, such as massive stars as demonstrated by the work of Chandrasekhar in 1983. General relativity shows that black holes are remarkably simple objects which are characterized by just a few numbers. As stated by Chandrasekhar the black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time (Hartle, 2003). The existence of black holes was first discussed by Michell and Laplace within the framework of the Newtonian theory at the end of the 18th century (Frolovp and Zelnikov, 2011). It was then viewed as a star with strong gravitational field so that the Newtonian escape velocity 2GM=R (with M and R being the mass and radius of the star respectively) is larger than the speed of light. In fact, the inequality R ≤ 2GM=C2 for escape velocity holds in general relativity (Penrose, 2004; Krishnan, 2013). Kerr discovered a solution of the Einstein equation, which describes the gravitational field of a stationary rotating black hole in 1963 (Jakobsson, 2017). This solution has a gravitational radius which prescribes the position of the event horizon (Frolov and Zelnikov, 2001). Carter explained its global properties in 1966 (Booth, 2005). The Kerr-Newman solution which represents charged spinning black holes was discovered in1965. John Wheeler introduced the term black hole in 1967 (Frolov and Zelnikov, 2011; Krishnan, 2012). There were seminal developments at that very helping to understand the general properties of black holes. These included the study of the global properties of black hole spacetimes, the definition of event horizon, the singularity theorems of Penrose and Hawking as well as the introduction of the concept of trapped surfaces by Penrose (Krishnan, 2012). The black hole uniqueness theorems which showed that the Kerr-Newman solutions are the unique globally stationary black hole solutions in the Einstein-Maxwell theory in four dimensions was established in the 1980s following the work of Israel, Carter and Robinson (Krishnan, 2013). In the 1980s, Robinson and Carter (Krishnan, 2013) established the uniqueness theorems of the Kerr Newman solutions for the description of the black holes of nature. This theorem states that; stationary axisymmetric solutions of Einstein’s equation for the vacuum, which have a smooth convex event horizon, are asymptotically flat and are non-singular outside of the horizon, are uniquely specified by the two parameters, the mass and the angular momentum and these two parameters only (Chandrasekhar, 1983). These theorems assert that, given a matter model, a static or a stationary black hole spacetime belongs necessarily to a specific class of spacetimes (in the vacuum case, they are Schwarzschild in the static regime and Kerr for the stationary case) which are characterized by a few parameters that describe the fundamental properties of the black hole. Thus, 24 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 10, No. 4; 2018 the Kerr solution represents the unique solution which the general theory of relativity provides for the description of all black holes that can occur in the astronomical universe by the gravitational collapse of stellar masses; and it is the only instance of a physical theory providing an exact description of a macroscopic object (Chandrasekhar, 1983). The study of black holes has for many years depended on event horizons as the boundary of the region of the black hole from where one can send signals to infinity (Senovilla, 2011). However, the study of black holes based on the concept of classical event horizon has the following drawbacks: to locate a black hole region and event horizon requires the knowledge of the entire spacetime and the definition has no direct relation with the notion of strong gravitational field as shown by (Ashtekar and Krishnan, 2004) and (Krishnan, 2013). For example in the Vaidya spacetime, event horizon can form in a flat region. Another global feature of event horizons is their teleological nature (Gourgoulhon and Jaramillo, 2008). The classical black hole boundary, i.e. the event horizon, responds in advance to what will happen in the future. Booth (2005) showed this using the explicit example of a black hole formed by the collapse of two successive matter shells: after the first shell has collapsed to form the event horizon, the latter remains stationary for a while and then starts to grow before the second collapsing shell reaches it (Gourgoulhon and Jaramillo, 2008). If black holes are considered as ”ordinary” physical objects, for instance in quantum gravity or numerical relativity, the above mentioned global behaviour of the event horizon would be problematic (Gourgoulhon and Jaramillo, 2008). The global nature of the event horizon and these physical problems associated with the event horizon have motivated the need to use a local approach as a complementary means of characterizing black holes. Through the Hawking and Penrose’s singularity theorems and weak cosmic censorship, the existence of a black hole region is indicated. In fact, in strongly predictable spacetimes satisfying proper energy conditions, trapped surfaces are guaranteed to lie inside the black hole region. Moreover; their location does not involve a whole future spacetime development (Jaramillo, 2011). The purpose of this paper is to explicitly demonstrate the existence of trapped surfaces and marginally trapped surfaces by computing the expansions of null vectors (Krishnan, 2003) in the Kerr black hole region. The plan of the paper is as follows: section 2 discusses the covariant divergence which is the main tool for the computation of the divergences of both ingoing and outgoing null vectors. We also discuss Gauss divergence theorem for the com- putation of the flux of a vector field. This section also contains discussions of the local characterizations of black holes. A general discussion of the Kerr black hole in Boyer-Lindquist coordinate systems are presented in the same section. In section 3, the existence of trapped and marginally trapped surfaces in the Kerr black hole is discussed after computing their covariant divergences. In this same section, we apply Gauss’ divergence theorem to compute the fluxes of vector fields to support the claim that trapped and marginally trapped surfaces exist in the Kerr black hole. This is the main result of this paper. Section 4 then gives the conclusion of the result. 2. Local Characterizations of Black Holes 2.1 Trapped Surface A two-dimensional surface S in a four dimensional spacetime has two null directions normal to the surface at each point. Trapped surfaces are characterized via the covariant divergences of such vectors which are orthogonal to the surface. We can thus distinguish two future directed null vectors emerging from the surface S. If we denote the in-going and out-going α α null normals to the surface S by l and n respectively, then θl and θn are their respective divergences. The surface S is said to be trapped if both divergences are negative: θl < 0 and θn < 0 or according to Penrose, a trapped surface S is a compact, space-like 2-dimensional sub-manifold of space-time on which θlθn > 0 (Ashtekar and Krishnan, 2004).In flat space, the out-going light rays diverge and the ingoing ones converge, i.e.θl > 0 and θn < 0 so trapped surface cannot exist there. The notion of trapped surfaces, due to Penrose (1965), entails that, in a sufficiently strong gravitational field, as in gravitational collapse, even outgoing light rays converge. In stationary black holes,such as the Schwarzschild black hole, the event horizon and the Killing horizon